Update index.html

Author: Gmac4247

index.html

@@ -3023,7 +3023,7 @@ <h3 itemprop="name" style="margin:7px">Circumference of a Circle</h3>
 <h4 itemprop="name" style="margin:12px">Archimedes and the Illusion of Limits</h4>
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-<p itemprop="description" style="margin:12px">The Greek Archimedes’ method for estimating the π is often celebrated as a foundational triumph of geometric reasoning.
+<p itemprop="description" style="margin:12px">The Greek Archimedes’ method for estimating the pi is often celebrated as a foundational triumph of geometric reasoning.
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@@ -3103,7 +3103,7 @@ <h4 itemprop="name" style="margin:12px">Archimedes and the Illusion of Limits</h
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-<p itemprop="description" style="margin:12px"><strong>Archimedes' area formula A = πr² is not a direct result of calculus. It’s reverse-engineered by multiplying the circumference formula C = 2πr by half the radius—treating the area as the sum of infinitesimal rings. While the result of that method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.</strong>
+<p itemprop="description" style="margin:12px"><strong>Archimedes' area formula A = pi × r² is not a direct result of calculus. It’s reverse-engineered by multiplying the circumference formula C = 2pi × r by half the radius—treating the area as the sum of infinitesimal rings. While the result of that method is algebraically valid, it bypasses the geometric logic that defines area: the comparison to a square.</strong>
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@@ -3120,7 +3120,7 @@ <h4 itemprop="name" style="margin:12px">The Symbol Pi: A Linguistic Shortcut</h4
 <strong>But this is not necessarily the ratio itself—it’s the notation of that ratio.</strong> That distinction matters. There was a ratio between circumference and diameter long before the Greeks studied it. We must not let their symbolic shortcut overwrite a more fundamental geometric truth.
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-It was not until the 18th century that the symbol π, popularized by the mathematicians of the time, gained widespread acceptance. 
+It was not until the 18th century that the symbol pi, popularized by the mathematicians of the time, gained widespread acceptance. 
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@@ -3153,7 +3153,7 @@ <h4 itemprop="name" style="margin:12px">∫ Calculus: Summary, Not Source</h4>
       <mn>0</mn>
       <mrow>
         <mn>2</mn>
-        <mi>&#x03C0;</mi> <!-- pi -->
+        <mi>pi</mi> 
       </mrow>
     </msubsup>
     <mi>r</mi>
@@ -3228,9 +3228,9 @@ <h4 itemprop="name" style="margin:12px">A Rational Alternative: 3.2</h4>
 Unfortunately, the exact details of the proposed method in the Indiana Pi Bill are somewhat obscure and have been interpreted differently by various accounts. 
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-The π has served its symbolic purpose. But in geometry, clarity matters more than tradition. 
+The pi has served its symbolic purpose. But in geometry, clarity matters more than tradition. 
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-The π is a fundamental constant in the geometry of idealized circles and plays a crucial role in many mathematical theories. 
+The pi is a fundamental constant in the geometry of idealized circles and plays a crucial role in many mathematical theories. 
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 However, using geometric construction and algebraic simplification we find that when we move from these idealizations to the measurement of real objects, a slightly different constant, 3.2 emerges as more relevant for accurately describing their properties.